List the elements of the set S={(x, y) ∈𝐙 ×𝐙:|x|+|y|=3}. Plot the corresponding points in the Eu

step 1 In this question, the question is saying that we have to describe set of all points by using the

List the elements of the set S={(x, y) ∈𝐙 ×𝐙:|x|+|y|=3}....

Key Concepts

Absolute Value

The absolute value of a number, denoted |x|, represents its distance from zero on the real number line, disregarding its sign. In equations involving absolute values, the expression |x| + |y| = constant defines a relationship based solely on the magnitude of x and y, which is essential in understanding how solutions can be derived without considering their sign directly.

Manhattan Distance

The equation |x| + |y| = 3 is a formula for Manhattan distance (or L1 norm), which measures distance as the sum of the absolute differences in the horizontal and vertical directions. This concept models scenarios where movement is restricted to grid-like paths, and the set of points satisfying this equation forms a 'diamond' shape centered at the origin in the coordinate plane.

Lattice Points

Lattice points are points in the Euclidean plane with integer coordinates. When we restrict our solutions to the equation |x| + |y| = 3 to integer values, we are finding all lattice points on the boundary of a diamond-shaped region. This restricts the set to discrete points, making enumeration and plotting straightforward.

Graphing in the Euclidean Plane

Plotting points in the Euclidean plane involves marking coordinates determined by their x and y values on a Cartesian grid. For the equation |x| + |y| = 3, each solution corresponds to a specific point that, when plotted, illustrates the geometric structure (a diamond) of the set of solutions. This visualization helps to bridge the gap between algebraic definitions and geometric interpretation.

Transcript

Please give Ace some feedback